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<div class="title">ConvAnalytical.py</div>  </div>
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<div class="fragment"><pre class="fragment"><a name="l00001"></a>00001 <span class="comment">#!/usr/bin/python</span>
<a name="l00002"></a>00002 <span class="comment"># -*- coding: utf-8 -*-</span>
<a name="l00003"></a>00003 <span class="keyword">from</span> sympy <span class="keyword">import</span> *
<a name="l00004"></a>00004 <span class="keyword">from</span> numpy <span class="keyword">import</span> array, zeros, matrix
<a name="l00005"></a>00005 
<a name="l00006"></a>00006 init_printing(use_unicode=<span class="keyword">False</span>, wrap_line=<span class="keyword">False</span>, no_global=<span class="keyword">True</span>)
<a name="l00007"></a>00007 x = Symbol(<span class="stringliteral">&#39;x&#39;</span>)
<a name="l00008"></a>00008 y = Symbol(<span class="stringliteral">&#39;y&#39;</span>)
<a name="l00009"></a>00009 
<a name="l00010"></a>00010 
<a name="l00011"></a>00011 <span class="comment"># Test cases. construct the exact solutions.</span>
<a name="l00012"></a>00012 
<a name="l00013"></a>00013 exact1 = 1
<a name="l00014"></a>00014 exactx = x
<a name="l00015"></a>00015 exactx2 = x**2
<a name="l00016"></a>00016 
<a name="l00017"></a>00017 
<a name="l00018"></a>00018 kernel = 1
<a name="l00019"></a>00019 
<a name="l00020"></a>00020 u1kx = integrate(exact1* kernel, (x, -1, 1))
<a name="l00021"></a>00021 uxkx = integrate(exactx* kernel, (x, -1, 1))
<a name="l00022"></a>00022 ux2kx = integrate(exactx2* kernel, (x, -1, 1))
<a name="l00023"></a>00023 
<a name="l00024"></a>00024 <span class="keywordflow">print</span> <span class="stringliteral">&#39;kernel 1 \n\n&#39;</span>
<a name="l00025"></a>00025 
<a name="l00026"></a>00026 <span class="keywordflow">print</span> <span class="stringliteral">&#39;u=1 | conv = &#39;</span>, u1kx
<a name="l00027"></a>00027 <span class="keywordflow">print</span> <span class="stringliteral">&#39;u = x | conv = &#39;</span>, uxkx
<a name="l00028"></a>00028 <span class="keywordflow">print</span> <span class="stringliteral">&#39;u = x^2 | conv = &#39;</span>, ux2kx
<a name="l00029"></a>00029 
<a name="l00030"></a>00030 <span class="keywordflow">print</span> <span class="stringliteral">&#39;\n\n&#39;</span>
<a name="l00031"></a>00031 
<a name="l00032"></a>00032 
<a name="l00033"></a>00033 kernel = y-x
<a name="l00034"></a>00034 
<a name="l00035"></a>00035 u1kx = integrate(exact1* kernel, (x, -1, 1))
<a name="l00036"></a>00036 uxkx = integrate(exactx* kernel, (x, -1, 1))
<a name="l00037"></a>00037 ux2kx = integrate(exactx2* kernel, (x, -1, 1))
<a name="l00038"></a>00038 
<a name="l00039"></a>00039 <span class="keywordflow">print</span> <span class="stringliteral">&#39;kernel x \n\n&#39;</span>
<a name="l00040"></a>00040 
<a name="l00041"></a>00041 <span class="keywordflow">print</span> <span class="stringliteral">&#39;u=1 | conv = &#39;</span>, u1kx
<a name="l00042"></a>00042 <span class="keywordflow">print</span> <span class="stringliteral">&#39;u = x | conv = &#39;</span>, uxkx
<a name="l00043"></a>00043 <span class="keywordflow">print</span> <span class="stringliteral">&#39;u = x^2 | conv = &#39;</span>, ux2kx
<a name="l00044"></a>00044 
<a name="l00045"></a>00045 
<a name="l00046"></a>00046 kernel = (y-x)**2
<a name="l00047"></a>00047 
<a name="l00048"></a>00048 u1kx2 = integrate(exact1* kernel, (x, -1, 1))
<a name="l00049"></a>00049 uxkx2 = integrate(exactx* kernel, (x, -1, 1))
<a name="l00050"></a>00050 ux2kx2 = integrate(exactx2* kernel, (x, -1, 1))
<a name="l00051"></a>00051 
<a name="l00052"></a>00052 <span class="keywordflow">print</span> <span class="stringliteral">&#39;\nkernel x** 2 \n\n&#39;</span>
<a name="l00053"></a>00053 <span class="keywordflow">print</span> <span class="stringliteral">&#39;u=1 | conv = &#39;</span>, u1kx2
<a name="l00054"></a>00054 <span class="keywordflow">print</span> <span class="stringliteral">&#39;u = x | conv = &#39;</span>, uxkx2
<a name="l00055"></a>00055 <span class="keywordflow">print</span> <span class="stringliteral">&#39;u = x^2 | conv = &#39;</span>, ux2kx2
<a name="l00056"></a>00056 
<a name="l00057"></a>00057 <span class="keywordflow">print</span> <span class="stringliteral">&#39;\n\n&#39;</span>
<a name="l00058"></a>00058 
<a name="l00059"></a>00059 
<a name="l00060"></a>00060 <span class="comment"># compute the matrix analytically.</span>
<a name="l00061"></a>00061 kernel = x - y
<a name="l00062"></a>00062 (gx0, gx1, gx2) = (-0.5 * x * (1 - x), 1. - x ** 2, 0.5 * x * (1 + x))
<a name="l00063"></a>00063 (fy0, fy1, fy2) = (-0.5 * y * (1 - y), 1. - y ** 2, 0.5 * y * (1 + y))
<a name="l00064"></a>00064 
<a name="l00065"></a>00065 f0 = integrate(fy0 * kernel, (y, -1, 1))
<a name="l00066"></a>00066 <span class="keywordflow">print</span> <span class="stringliteral">&quot;\Phi_0 convolve against 1_[-&quot;</span>+str(1)+<span class="stringliteral">&quot;,&quot;</span>+str(1)+<span class="stringliteral">&quot;] &quot;</span>, f0
<a name="l00067"></a>00067 f1 = integrate(fy1 * kernel, (y, -1, 1))
<a name="l00068"></a>00068 <span class="keywordflow">print</span> <span class="stringliteral">&quot;\Phi_1 convolve against 1_[-&quot;</span>+str(1)+<span class="stringliteral">&quot;,&quot;</span>+str(1)+<span class="stringliteral">&quot;] &quot;</span>, f1
<a name="l00069"></a>00069 f2 = integrate(fy2 * kernel, (y, -1, 1))
<a name="l00070"></a>00070 <span class="keywordflow">print</span> <span class="stringliteral">&quot;\Phi_2 convolve against 1_[-&quot;</span>+str(1)+<span class="stringliteral">&quot;,&quot;</span>+str(1)+<span class="stringliteral">&quot;] &quot;</span>, f2
<a name="l00071"></a>00071 
<a name="l00072"></a>00072 a = matrix(zeros((3, 3), <span class="stringliteral">&#39;float&#39;</span>))
<a name="l00073"></a>00073 a[0, 0] = integrate(gx0 * integrate(fy0 * kernel, (y, -1, 1)), (x, -1, 1))
<a name="l00074"></a>00074 a[0, 1] = integrate(gx0 * integrate(fy1 * kernel, (y, -1, 1)), (x, -1, 1))
<a name="l00075"></a>00075 a[0, 2] = integrate(gx0 * integrate(fy2 * kernel, (y, -1, 1)), (x, -1, 1))
<a name="l00076"></a>00076 
<a name="l00077"></a>00077 a[1, 0] = integrate(gx1 * integrate(fy0 * kernel, (y, -1, 1)), (x, -1, 1))
<a name="l00078"></a>00078 a[1, 1] = integrate(gx1 * integrate(fy1 * kernel, (y, -1, 1)), (x, -1, 1))
<a name="l00079"></a>00079 a[1, 2] = integrate(gx1 * integrate(fy2 * kernel, (y, -1, 1)), (x, -1, 1))
<a name="l00080"></a>00080 
<a name="l00081"></a>00081 a[2, 0] = integrate(gx2 * integrate(fy0 * kernel, (y, -1, 1)), (x, -1, 1))
<a name="l00082"></a>00082 a[2, 1] = integrate(gx2 * integrate(fy1 * kernel, (y, -1, 1)), (x, -1, 1))
<a name="l00083"></a>00083 a[2, 2] = integrate(gx2 * integrate(fy2 * kernel, (y, -1, 1)), (x, -1, 1))
<a name="l00084"></a>00084 
<a name="l00085"></a>00085 rhs = array([[1.], [1.], [1.]])
<a name="l00086"></a>00086 
<a name="l00087"></a>00087 <span class="keywordflow">print</span> rhs
<a name="l00088"></a>00088 <span class="keywordflow">print</span> a
<a name="l00089"></a>00089 
<a name="l00090"></a>00090 conv = a * rhs
<a name="l00091"></a>00091 <span class="keywordflow">print</span> conv
<a name="l00092"></a>00092 
<a name="l00093"></a>00093 <span class="keywordflow">print</span> conv[0] * f0 + conv[1] * f1 + conv[2] * f2
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